We made plots for five different cases: \(\xi =\lbrace -0.4,-0.2,0.0,0.2,0.4\rbrace \) . The plots of the scalar field \(f\)  and the gauge field \(a\)  all plateau at unity regardless of \(\xi \) . We also plot the magnetic field whose profile depends on \(\xi \) . The most important plot by far is the one for the metric \(A\)  which plateaus asymptotically to a constant value (which we previously labeled \(D\) ). The profile of the metric here (starting at unity at the origin and then plateauing to \(0<D<1\) ) is in stark contrast to the AdS\(_3\)  case where the metric had an \(r^2\)  dependence asymptotically. The constant \(D\)  can only be obtained numerically (by running the simulation) and it changes with \(\xi \) . Since the deficit angle depends on \(D\)  via (REF ), the deficit angle depends on \(\xi \) . We also calculate the mass \(M_{flat}\)  of the vortex via (REF ). The constant \(D\) , the deficit angle \(\delta \) , the mass \(M_{flat}\)  as well as the peak value of the magnetic field are presented in table REF . In \(2+1\) -dimensional General Relativity in asymptotically Minkowski spacetime, there is the classic result due to Deser et al. [1]} that a point mass produces a deficit angle proportional to the mass. The ratio of mass to deficit angle is equal to \(2\ \alpha =1/(8\,\pi \,G)\)  and is a constant since Newton's constant \(G\)  does not change as the mass changes. In contrast, for the vortex with non-minimal coupling, the ratio of mass to deficit angle is not constant but depends on \(\xi \) : it is equal to \(2\alpha _{eff}=2(\alpha +v^2\,\xi )=2\,(1+\xi )\)  where we substituted the values \(\alpha =1\)  and \(v=1\)  for our parameters. A striking consequence is that it is possible for a larger mass to actually produce a smaller deficit angle compared to a smaller mass. For example, in table REF , the case at \(\xi =-0.4\)  has the largest deficit angle of 1.894 rad in our sample and has a mass of 2.273 whereas the case at \(\xi =0.4\)  has a smaller deficit angle of 1.225 rad but the largest mass of
3.430 in our sample which is roughly 1.5 times greater than our former case.
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